a Horizontal Layer of Fluid. 539 



If a:, or v, vanish there is instability ; but if tc and v are 

 finite and large enough, the equilibrium for this disturbance 

 is stable, although the higher temperature is underneath. 



Inequality (39) gives the condition of instability for the 

 particular disturbance (I, m, s). It is of interest to inquire 

 at what point the equilibrium becomes unstable when there 

 is no restriction upon the value of V 2 + m 2 . In the equation 



P'y(P + m*)-Kvo 3 = p'y(<r- s*)-ki>o* = 0, . (40) 



we see that the left-hand member is negative when l 2 + m 2 

 is small and also when it is large. When the conditions 

 are such that the equation can only just be satisfied with 

 some value of l 2 + m 2 t or cr, the derived equation 



/3' 7 -3/cv(7 2 = (41) 



must also hold good, so that 



o-=3s 2 /2 J Z 2 + wi 2 =4* s , . . . (42) 



and /3' 7 =27,n</4 (43) 



Unless fi'y exceeds the value given in (43) there is no 

 instability, however I and m are chosen. But tie equation 

 still contains s, which may be as large as we please. The 

 smallest value of 5 is 7r/f. The condition of instability 

 when /, m, and s are all unrestricted is accordingly 



^7>-4^ ■ (44) 



If fi'y falls below this amount, the equilibrium is altogether 

 stable. I am not aware that the possibility of complete 

 stability under such circumstances has been contemplated. 



To interpret (44) more conveniently, we may replace fi' 

 by (©a-00/f and y by gfa-pd/pifa-Qfi, so that 



0V=f&=a f (45) 



S Pi 



where ® 2 j ©i> Pit and p 1 are the extreme temperatures and 

 densities in equilibrium. Thus (44) becomes 



(^>*££ (46) 



Pi ±9? v ; 



In the case of air at atmospheric conditions we may take 

 in C.G.S. measure 



v = -14, and «=|v (Maxwell's Theory). 



